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Electroproduction of strange particles at high energies
Nuclear Physics 69 (1965) 637-651; Not
to be
reproduced
ELECTROPRODUCTION
by
photoprint
@ North-Holland Publishing Co., Amsterdam
or microfilm
OF STRANGE
without
written
PARTICLES
permission
AT HIGH
from
the
publisher
ENERGIES
M. P. REKALO Physical-Technical
Institute of the Ukrainian Academy of Sciences, Kharkov, USSR
Received 23 June 1964 Abstract: The K meson electroproduction amplitude at high energies is obtained on the basis of the Regge pole hypothesis in the region of 180”. Two possible variants of the internal parity of the system (KNY) are considered. The asymptotic amplitudes are represented in the factorized form corresponding to the contribution of two Feynman graphs describing the exchange of a reggeon. The helical amplitudes which are oscillating functions of the energy and angle of production are calculated; the contribution of the longitudinal polarization of a virtual photon proves asymptotically smaller than the transverse polarization contribution. Different observed characteristics of the electroproduction of K mesons are calculated.
1. Introduction The investigation of the analytical properties of partial-wave amplitudes as functions of the angular momentum has proved extremely effective in exploring the asymptotic behaviour of the amplitudes of different processes at high energies ‘). The simplest hypothesis based on the Mandelstam representation and two-particle unitarity has been considered very intensely in quantum field theory. According to this hypothesis the scattering amplitudes at high energies are only determined by the simple poles of the partial-wave amplitude. The investigation of the many-particle terms of the unitarity condition recently *, “) undertaken shows that the analytical properties of partial-wave amplitudes are not as simple as was first thought; in addition to moving poles there are also moving branching points. However, many features of the purely pole asymptotic behaviour hold good in the case of a more complicated picture of analytical properties in the complex j plane. In particular, there remain the spin structure of asymptotic amplitudes, factorizing and isobaric relations and the oscillation behaviour of large angle scattering amplitudes. The production of strange particles in inelastic electron-nucleon scattering at high energies in the angular domain where the asymptotic behaviour of the amplitudes is determined by fermion Regge poles is considered in this paper. The investigation of this process is of double interest; on the one hand, the electroproduction of K mesons is an elementary process of production of strange particles and on the other hand this process possesses some interesting peculiarities making it different from the 637
638
M.
production
of K mesons
P.
REKALO
by real y quanta.
First, the square
of the four-momentum
of a virtual photon is not zero; secondly, besides the transverse polarization the virtual photon has longitudinal polarization; thirdly, the differential cross section for the electroproduction of K mesons averaged over the polarization of all particles involved contains a term corresponding to the photoproduction of K mesons by a linearly
polarized
photon.
by non-polarized
Therefore,
electrons
the investigation
the investigation
at high energies
of the photoproduction
of the production
may furnish
of K mesons
of K mesons
the same information by polarized
as
real y quanta.
2. Structure of the Amplitude in the S-channel The
K
determined
meson-nucleon
electroproduction
amplitude
(e- + N + e- + Y + K)
is
as follows: si,
=
6if_i64(rl+P1-r2-P2-q)
mZ%M2
A,
2wq .QE, E2
(274%
(1)
where Sif is the S-matrix element, rl , m, El (r2, m, E2) are the momentum, mass and energy of the initial (final) electron, pl, M, and E, are the four-momentum, mass and energy of the nucleon, p2, M2 and E, the four-momentum, mass and energy of the hyperon, and 4, w are the momentum and energy of the K meson. If the electromagnetic interaction is taken into account in the lowest order in perturbation theory, we obtain the amplitude
A=
where k2 = (rl -~2)~ interacting particles. The quantity
and J,
is the electromagnetic
EJ is expanded
EJ =
in independently
5Ai(s,t,
~2
current
invariant
operator
of strongly
combinations
k2)Mi(E, k, ~1) p2),
i=l
M,
=
rsR,
M4 = -il3,
M2
=
rEP2,
M,
M,
= TEP1,
s = -(k+pd2,
where r = iys if the internal it is positive.
f2%(P24lJ,lP,h
= -iT&p21,
b!fe = -irEp,&
u = -(k-p2)2,
parity
(2)
of the system
t = -(pi
(KNY)
-P~)~,
is negative
and P = 1 of
639
STRANGE PARTICLES
3. Structure of the Amplitudes in the U-channel; KNY Parity Negative Let us first consider the case of negative parity. Since in the following we shall be interested in the asymptotic behaviour of electroproduction amplitudes, s --) co, u = const, t + -co, which corresponds to the production of K mesons at 180” in the c.m.s. of the K meson and hyperon,
it is necessary
procedure,
in the u channel
the partial-wave
y* is a virtual
photon.
expansions
To realize the transition
make the substitution k -+ -k in eq. (2). In eq. (2) we eliminate the time component conservation equation k,fi(0+,44 and pass amplitude 9,
to two-component
= ia - 8Flu(w,, x F3u(w,,
Then
we obtain
cos
e,) =
87cw,
92u(w,, cos e,)
sqw,,
cos
. ii8 . IW~~(W,,
cos
Thu(wu, cos
ia
. ;E
cos e,)
. LF6u(~,,
amplitudes
cos e,),
as follows:
_M2)_A4],
U
=
e,) = (E,,+M,)J(E,,-M,)(E,“-Mz) 8nw,
~4u(wtl,cos
current
for the two-component
with the invariant
J&.+w)(E2u+m[_qw
to
cos 0,) + ia * 8%~ . tj
+
sqw,,
where
it is necessary
= 0
cos 0,) + ia . @ . Q3F4u(w,, cos Q,) + ia
are connected
+ Y +y*),
eP with the aid of the electron
cos 0,) + ia * ka . EC * tjFzu(w,,
where the scalar amplitudes
by the conventional
(K+N
into the u channel
= Q,,
quantities.
to obtain,
(j ) = (E,,-Ml)~\I(El,+M,)(E,,+M,) u 871~~
0,) =
0,) =
CA3_(W +M2)A6~, U
[-A3_(W
u _M,)‘&],
(E,,-M,)J(E,,+M,)(E,,+Mz) 8j-w,kou
(E2,+M,)J(E,,-M,)(E,,-M,) 8nwu
[_A,cw kou
_M2j_A, u
(3)
640
P. REKALO
M.
The kinematic
coefficients E
here are functions
_ n+W-P2 lu 247
E2u =
’
u-M;-k2 ko, = .It is necessary
of U: tr+M;+k’ 2Ju
w, = Jzl.
2Ju
’
to note that the quantity
k2 (the square of the virtual photon
is assumed to be a finite quantity, which corresponds angles. The invariant amplitudes satisfy under our assumption tion. Hence from eq. (3) follow the symmetry properties 91.(M’,,
= -2
f5 s (-+o~LF.~+o)
the Mandelstam
cos O,),
9-3U(ll?U, cos e,> = -P4u(-M’,,
cos O,),
Let us then introduce
sin 30,
= -2
f6 E (+opF”l*o) = 2 ~0s
helical amplitudes:
Cf:‘(~.)Cpg+;(z.)+p;-~(~.)l,
sin ie,
C~~~(W,)[PJ+)(Z,)+PJ-~(Z,)I,
*e, ~fa’;~.)cp;~:(~,)-P;-~(3.)1. .i
?I = cos e,, connected
with the scalar amplitudes fi = -&/2
by the relations
sin ~eu[2(~,,+~2u)+(i+c~~
f2 = -:JZ cos :e,[2(9-,,-
s2J
e,)(9,,+94Uyj, - (I-
f3 = $42 cos *e,(i - cos e,)(F,,
- Fdu),
f4 = *JZ sin :e,(i + cos e,)(fl,,
+ FJ,
fS = -sin
~eu(~lu-~2u+c0s
f6 = -C~S ~e,(~,,+3=2u+c0s
cos e,w,,-
representa-
(4)
cos 0,) = - P-eu( - W,) cos 0,).
the following
“mass”)
to small electron-scattering
cos Q,) = -92U(-lIt’,,
SJW,)
fi 5 (fllF,l*o)
-’
=cu)it
e,93u-cos
e,~~,+9,,-p,,),
e,9-3u+cos
e,~,,+~,,+9-,,).
STRANGE
If we introduce
the partial
641
PARTICLES
wave amplitudes
with a definite
parity
with the aid of
the relations f/
= i(h( + hi),
f;’ = +(/ii -h$, we obtain according
j-3’ = +(hj+h$
fsj = *(hi + hj,),
fi = #j-h’,),
fJ = +(h< -hi),
to eq. (4) h{(w,,) = -hi(-w,), h’3(WU)= hj4( - WJ,
(6)
h<(w,) = hj6( - WJ. These relations are central in the theory of fermion Regge poles. It follows from them that if, say h{(w,) has a certain singularity in the j plane and its position is determined by the condition j = a(~,), then h{(wJ also has a singularity in the j plane at the point j = a( - w,). If the singularity is a pole, the subtraction of the amplitude h{(w,) at the polej = LY(W,)and the subtraction of the amplitude hi(~v,) at the polej = CX(- wJ are interconnected. Assuming that c((EJ,) is real in the region of bound states, Gribov “) has shown that Im c((w,) # 0 at u < 0 i.e., the amplitudes have an oscillation character at high energies and scattering angle M 180”.
4. Asymptotic
Invariant Amplitudes;
KNY Parity Negative
With the aid of the Sommerfeld-Watson transformation we pass in eq. (5) from summation in j to integrals, deforming the integration contour in the complex region by extending it from a- ice to a+ ice. Assuming that to the right of the straight line Rej = a there lie only poles of partial-wave amplitudes we obtain asymptotically for z, + 00, u = const, corresponding to high energies in the s channel and K meson production angles z 180”, the following expressions, taking into account the conjugation of the trajectories of fermion poles with opposite parity: -+ f1 2 sin +e, ___fi 2 sin :e,
f3 2 cog +e,
j-2 = Q;(&))*
2 cos +e, -
+
~ fi 2 cos :e,
f4
------_=
= -2&Ju)
_2(x;(Ju))*
2 sin +e,
- ~ f4 = -2&/u) 2 sin +e, 2 cos +e, = 2(x:(Ju))* 2 cog 3e,
,i-3f(_s)j-* cos zj
’
sj*-w-#‘-+
)
cos nj*
f3
_+Lf5 2 sin $e,
sj*-++,i._, )
si-3+(_s)i-+, cos nj ~j*-++(-sY*--t cos nj*
)
642
M.
L
fs
-
2 sin *Q,
____
f6
P.
=
REKALO
,j-t+(_,)j-+,
-2&/u)
2 cos +e,
cos rcj
where x1, x2 and x3 are the subtractions II{, h< and hi multiplied by a certain function of u originating in the consideration of the asymptotic behaviour of the Legendre polynomials Using
and the sign (+ ) corresponds
these
expressions
to different
let us determine
the
signatures.
asymptotic
forms
for
invariant
amplitudes Al(s,
u)
=
sj-++p
ul(Ju)
~i*-3f(-sY*-+)
+uy(Ju)
cos 2-cj A,($
+
=
cos
nj*
p1 (I- $) +a,(Ji+M,)] q-~~;y-+
+ [“T (I+ 5) -u;(:u--rM2)]s-y*-+
)
-a,(Ji-M,)
A, =
,j-t*(_,)j-P
sj*-++s)j*-*,
+uT(Ji+M2)
cos nj
A,(s,u) =
(a, +a, Ju
,j-+*(__,)i-+
1 -
cos nj
(7)
cos nj" -
(Ju- 21 *
u;
,j*-l+(_,)j*-t
--a
9
cos nj*
where
‘I (“)
________--4rtJ2 _~ = J(E,,-M,)(E,,-M,)
a2
=
(x: -xl),
4i+k,, (44
JU(E2,+M2)VI(E,.--M,)(E,,~
(xl’
+x:
+JW
The amplitudes A, and A, are asymptotically small. Thus, the invariant amplitudes are determined by the two quantities al and ~2, the asymptotic structure of these amplitudes being such that the total amplitude of the virtual photoproduction of K mesons can be represented as follows: A = A(+)(& u)+P(s,
u),
A(‘)(&u) = u(p,)B[b,(Ju)+il;b,(Ju)](iJ’-Ju)y,u(p,)
,j-+*(_,)j-+
~
cos
A’-‘(S, U) = C(P~)J?[~:(~U) + iLbz(j~)J($+
nj
sj*-++(_s).i*-t Ju)Y~ U(pl) Lyp cos nj*
3
3
(8)
STRANGE
where bl and b, are determined
PARTICLES
by a, and a2 with the aid of the relations b,+b,(&+M,)
2b,-b&h+MJ
= -a,,
= -a,
l(
Eq. (8) corresponds Feynman
643
to the amplitude
3 +a,(Ju+M,). Ju 1
determined
by the contribution
of two pole
graphs. These graphs describe the exchange of a reggeon between the photon
and hyperon, on the one hand, and the K meson and nucleon, on the other; the to the propagation of the reggeon between the quantities ($+J u ) correspond vertices on the graphs, yS corresponds to the K + N -+ reggeon vertex and &(b, + if&,). the reggeon, to the Y +y* vertex. One graph describes the exchange of a reggeon with mass ,/u and spin j(Ju) and the other the exchange of a reggeon with mass -Ju
and spin j*(Ju). 5. Asymptotic
Spiral Amplitudes;
(KNY) Parity Negative
In the calculation of the differential cross sections and the quantities characterizing the different polarization effects in the K meson electroproduction, it is convenient to use the helical amplitudes in the s channel. To calculate these amplitudes in the c.m.s. of the s channel we introduce the two-component amplitude 9,
= ia.&~l,+ib.4u.Eb.12~2,+ib.l;&.q^~3S
+ia.de.4~,,+i6.1;E.f~c~+iu.~&‘1;~6s, where gIs
=
4 El~+“JE2~+!!!,)
[A,(W
s
-M,)-A,],
8?TW,
%F25 =
J(Els-~JE2s-~4
+M,)_A
[_Al(w
4
s
]
9
&CW,
s
3s
=
.
(E2,+~2)~(E1~-“,)02~-M2)~-_~2+(w.+~l)~,], _________
s
87CW,
F4,
=
(E,,-M,)J(E,,+M,)(E,,+M2) LA
+(w
_M
2
I
s
)A
,
5
9
8nw, .F
(E,,-IM,)J(E,,+M~)(E~~+M~) 5,
kos
(El,+M,)4(E,,-M,)(E,,--M,) 6s
+M,)_A s
8nWs
9
C_Al(w
=
[A,(w
_M,)_A, s
= 8nWs
kos
4
(9
644
M.
P.
REKALO
while s+Mf+k2 w, = Js, Using obtain
EI, =
relations
E
2Js
’
2b
(9) and asymptotic
s-M:-k2
= s+M,2-~’ 24s
expression
kos =
)
(7) for invariant
2Js
a
amplitudes
we
for helical amplitudes 1
($)l~sl+l)
=
I
S’--r*(-SY-r
r&/u)
sj*-+*(-s)i*-3)
+(r&/u))*
cos nj (+q~&l_+_l)
=
+(Ju)
cos xj*
si-i~(-4’-t
~j*-%-~~*-3)
+;(&))*
cos nj (g)lgq_g)
=
irF(Ju)
sj-w-4-+
cos nj*
f*-**(-sY*-3) (10)
+:(Ju))*
cos nj*
cos nj ($)l$zT
It-l)
=
@(Ju)
sj-t-*(-s)J-++($(&))*
sj*-w,:‘i”,
cos zj where
r:(Ju) = rz+(Ju)
=
~_ Ju
(xl’+x:+$x:)]
-xl’+xz’-
9
2&4u-Md(E,u-M,) ____Ju
xl’_x;-
2~(-4,-M,)(&,-MJ
2
:M (xl’+x:+$x:)] *
E 2u
2
The helical amplitudes corresponding to the longitudinal polarization of the photon are asymptotically smaller than those corresponding to its transverse polarization and therefore the longitudinal polarization contribution can be neglected.
6. Asymptotic
Invariant Amplitudes;
(KNY) Parity Positive
Let us now consider the case of positive parity of the system (KNY). of the u channel the two-component amplitude has the form Fu
In the c.m.s.
= 0 * La * E9=lu(W, ) cos 0,) + d - 86 * 4^92u(wu ) cos e,> +s * cj~3u(w” ) cos e,>
+ CJ* I;a * $5 . 4LF.+u(w. ) cos e,> + E . dF5u(w, ) cos e,> + d * ka * lj& * w6u(w,
) cos O,),
where 9
= lu
J(Eh‘+Ml)(E2U+M2) 8nw,
[_A,(w
11
+M
)_A 2
] 4
3
(11)
STRANGE
F
3u
PARTICLES
645
d&.+~d(E,u+~,)
=
h43-bJ”--&l~
87CW”
95” =
kuJ(EI”+wE2u+~2) 8nwu kou [A,(w
)_A
_M
4
2
”
+W,A,+E,,A,-W,(W,-M,)A,-E,,(W,-M,)A,I,
9
6u
k,Jch”+wE2U+w
=
+Mz)_A4
[_A,(w u
8nWu
kou
-WuA,-E,,A,-w,(W,+M,)A,-E,,(W,-M,)A,l.
From
relations
(11) there follow 9&(w, 93u(w,,
the symmetry
properties
for scalar
amplitudes
) cos e,> = - 9-2”( - w, ) cos O,), cos e,> = -T&(-w,
) cos S,),
F5u(w, ) cos e,> = - T6”( - W,) cos f3”). Repeating the arguments of the previous sections we obtain the following expressions for the invariant amplitudes in the asymptotic region s -+ co, u = const < 0:
A,(%u>= 4/u> A,(&
u)
=
-
+u;*(~u)~j*-,+(-s)‘*-f
,j-+*(-s)j-+
3
cos
zj
cos nj*
(a;+O;)si-i+(-S~-f ( M, ) Ju 1+
cos nj
A,(s, u) = a;(J;_M,)
l(
3 @;+a;)* 1 44
sj-%(-+Y-*
_a;*(J;+M,)
,j*-Q(-s)i*-+, cos nj* sj”-+*(-Vi
cos nj a;_a; A,(& u) = ___ Ju
,i-+k(_,)j-* cos nj
@)
cos nj* + (a;-a;)* Ju
~~*-+*(-s)j*-+ cos nj*
.
As to A3 and A, they are asymptotically small and a, and a2 are connected with the subtractions of the partial-wave amplitudes. This structure of the amplitudes in the asymptotic region makes it possible to represent the K meson electroproduction amplitude as the sum of contributions of
646
M. P. REKALO
two Feynman
graphs
whose character A =
was discussed
above:
A(+)(& u)+P(s,
u),
sj-++sy--t (13)
A’+‘(s,24)= Zi(p,)&[b;(Ju)+i~b;(Ju)](~-Ju)u(p,)
cos rj
)
,j*-+*(_,)j*-t
A’-+, 24)= u(p,)8[b;*(Ju)+i~b;*(Ju)](lf+Ju)u(p,) while the following
connection
3
cos nj*
holds:
zqju)
= (
b&/n)
1+ 3 +(a; +a;), Jn 1 .
= a!? u
The amplitude (13) in the case of positive parity of the system (KNY) differs from the amplitude (8) in that a unit matrix instead of yS corresponds to the K+N + reggeon vertex. In terms of the internal parity (KNY) this substitution seems quite natural. 7. Asymptotic
Spiral Amplitudes;
(KNY)
Let us calculate the asymptotic helical amplitudes for the positive parity variant (KNY). We proceed plitude
Parity Positive
in the c.m.s. of the s channel from the two-component am-
s +M,)-A,],
Yzs =
4&5+~1)(E2s442)
CA,(w
_M
F3s
=
9sJ(E1s+w)(E2s+~2)
CA
1
4,
1
+(w
2
8nw,
)_A
s
87cw,
_M
s
)A
1
]
5 3
STRANGE
9
5s
kJ&s+wEzs+M2)
=
Bnw,
du-
6s
PARTICLES
647
[Al(w
s
_M
kJ@ls-w)(E2s-w [_qw
=
1
)_A
4
kos
)_A
+M
s
1
4
8nwskos
-E,,A,-~,A~+E~~(w,-M~)A~+wS(W,+M~)A~I. Using the asymptotic amplitudes
@)jgsl
-3_
1) =
invariant amplitudes (13) we then obtain for the helical
_
$(Ju)
sj-+;+y-+
sj*-+i(-f*-’ ,
++;(Ju))*
cos (g)]
+~sl_+l)
=
+gs,+_l)
=
p;(JU)Sj-++wf
(15)
sj*-++w-,
_ir~(Ju)s’-+f(-sY-k++.;(&))* cos nj
($0,
7q
cos nj*
si*-3+c-r+)
+($(&))*
cos rcj
cos nj*
where r1 and r2 are connected as follows with the quantities a; and ai determining the factorized amplitudes r,(Ju)
= -I!!!5
r,(Ju)
= - -Yf!f-
G-al
8rrJ2
, Ju+M,
ju_M,
T
8x,/2
The helical amplitudes corresponding to the longitudinal polarizations of a photon are asymptotically small. 8. Characteristics
of Electroproduction
Using eqs. (10) and (15), we can determine different characteristics of the electroproduction of strange particles e.g., cross section, polarization hyperons and asymmetry. To this end let us first re-write eq. (1) as
si, =
e
2.Y
kZ (2~)~
2kos
---A(y*+N e1c2
-+ Y+K),
(16)
where A (Y*+ N + Y + K) is the matrix element of virtual photoproduction in which the quantity Ey,u gives the polarization and the factor J2k,, originates from the normalization.
M.
648
Then
the differential
can be represented do
=
cross section
REKALO
for the electroproduction
of strange
particles
as follows: (kp,)* + k*Ivff
a ko,m2 rlpl TC*elc2k4
where J(r,
P.
(rIpI)*-m*Mf
kp,
p2)* - m2 Mf/r,p,
da(y* + N + Y + K)d3r2,
is the relative velocity
of the initial electron
(17)
and the
of the nucleon of the target and J(kp,)*+k*Mf/kp, is the relative virtual photon and nucleon introduced in order to go over from (A(y* + N -+ Y + K)I* “velocity”
to da(y*+N -+ Y+K). After the averaging over the initial
and summing
over the final electron
spin states
we obtain do(y*+N
--f Y+K)
= 4& s
s,fi f,s don,
where I%’is the unit vector orthogonal to k and located in the plane produced by the vectors k, rl. In eq. (18) the term J,,f,, corresponds to the differential photoproduction cross section averaged over the virtual photon polarization. The term d~&J& corresponds to the cross section for the production of K mesons by transverse linearly polarized photons, 6,&j& is responsible for the production of K mesons by longitudinally polarized virtual photons, and (k,& + I%:k,)f,, describes the contribution due to the interference of the amplitudes for the production of K mesons by transverse and longitudinal virtual photons. Since the contribution from the longitudinal polarization of photons is asymptotically small, eq. (18) can be re-written as da(y*+N
-+ Y+K)/dQ
k2 de = smz dn +
(19
are the differential cross sections for the production where do/dQ and (da/dQ)(e,) of K mesons by non-polarized and linearly polarized transverse photons, and the axis x is directed along the vector k’. For the polarization of recoil hyperon we obtain in a similar manner p da(y*+N
+ Y+K) dQ
where P’ is the polarization vector when the photons the same vector when the photons are linearly polarized.
are non-polarized
and
P”
STRANGE
649
PARTICLES
For the azimuthal asymmetry of the production of K mesons on polarized nucleons, we have A do(y*+N
+ Y+K) dS2
9. Asymptotic Expressions For Characteristics of Electroproduction of Strange Particles Let us give the explicit asymptotic expressions for different quantities in eqs. (19)-(21), considering simultaneously both internal parity variants: do = ~(1+~:)(p12+p:2)s2j’-1, dS2 -
gje.) = g
+cos @(l-c&) cos (q,: -cp:)p:p;s2”-l,
(22)
where r:(Ju)
=
pfP’*,
r;(~u)
=
P;eiq2*,
a: =
chd’+slw’ ch nj” T sin 7cj’
and @ is the azimuthal angle. These quantities do not depend on the internal parity (KNY). For the polarization of recoil hyperons we have
P;!ti(e,) = cos@P;.g -cos@R, -COS
Py g
(e,) = sin @R, -sin3@R,,
3@R,,
P;!= 0,
where
R, = *(cc, sin fl)~:~s’j’-~,
R, = $(a,sin fi)p:‘s2”-r,
tg/?=-.
sh nj” cos ?rj’
The plus sign in [ &-] corresponds to the negative parity of the system KNY and the minus sign to the positive parity. The system of axes x’, y’, z’ is connected with the recoil hyperon, being directed along the momentum of the hyperon.
650
M.
The asymmetry has the following
of the production
K.
REKALO
of K mesons
form when the photons
on polarized
nucleons
of the target
are non-polarized:
A;$ = t(l+~“,)(p:“+p:z)s2j’-l,
and the following
do A: dn = [i-]a+
sin /?(p:2+pt2)
sin @s2j’-i,
Ai g
sin /I(pf2+pi2)
cos @s2j’-i,
= [T]a*
form when the photons
are polarized:
A:’ $
(e,) = Ai g
+(sin
Ai’ $
(e,) = ‘4; $
+ (cos @ - cos 3@)C,
C = [-k]x*
sin bp:pz
@+sin 3@)C,
sin (cpf -‘p:)s2j’-l.
Contrary to the oscillatory behaviour of helical amplitudes all the above quantities are monotonic functions of the energy and angle. Only those quantities which characterize the polarization of recoil hyperons versus the polarization of the target nucleons are oscillatory, e.g. TzY,$ Q, = [r]+(~+
(4
=
Q1 +Q2,
Q2 = -4 h (fisfZse-iQ)+f3*sf4se3iQ),
sin p)(p:2+p~2)s2j’-1,
Im_flsf,*, = Imf3J4*, = -C, Refdi*,
= [&IP:P:[c&
sin (j”i+qp1kP)
cos (j”i+& -sin
Re_LJ,*,
=
[+ldd[ cc+ ? sin
(j”i+qP:*B)
cos (j”~+~P:)]s2j’-r,
cos (j”i+cp:+fl) -sin
where i = In sJsO and s,, is a certain squared.
(j”[+qi)
fP)
quantity
(j”[ + cpi) cos (j”c+ with the dimension
(p2+)]szi’-r, of an energy
10. Conclusion In conclusion let us formulate briefly the main results of this paper. (1) The amplitude of electroproduction of strange particles for asymptotically high energies and K meson production angles w 180” in the c.m.s. of final strongly
STRANGE
interacting
particles
can be represented
651
PARTICLES
in the factorized
form (8) or (13) depending
on the internal parity (KNY). This structure of the amplitude corresponds to the specific Feynman graphs describing the exchange of a reggeon. (2) The helical amplitudes in the asymptotic region are oscillating functions of the energy and angle; these amplitudes have a simple structure, which facilitates calculation of different characteristics of the process under study. (3) The contribution
of longitudinal
photons
is asymptotically
smaller
than
the the
contribution of transverse virtual photons. (4) Only the polarizations of recoil hyperons versus the polarization of target nucleons are observed oscillating quantities. All the other quantities are monotonic functions of the energy and angle. The author results.
wishes to thank A. I. Akhiezer
and D. V. Volkov for discussion
of the
References 1) V. N. Gribov, JETP 41 (1961) 66, 1962; S. C. Frautschi, M. Gell-Mann and F. Zachariasen, Phys. Rev. 126 (1962) 2204; S. C. Frautschi, G. F. Chew and S. Mandelstam, Phys. Rev. 126 (1962) 1202 2) S. Mandelstam, Nuovo Cim. 30 (1963) 1113, 1127, 1148 3) V. N. Gribov, I. Ya. Pomeranchuk and K. A. Ter-Martirosyan, Preprint, No. 222 Inst. for Theor. and Exp. Physics 4) V. N. Gribov, JETP 43 (1962) 1529
to be
reproduced
ELECTROPRODUCTION
by
photoprint
@ North-Holland Publishing Co., Amsterdam
or microfilm
OF STRANGE
without
written
PARTICLES
permission
AT HIGH
from
the
publisher
ENERGIES
M. P. REKALO Physical-Technical
Institute of the Ukrainian Academy of Sciences, Kharkov, USSR
Received 23 June 1964 Abstract: The K meson electroproduction amplitude at high energies is obtained on the basis of the Regge pole hypothesis in the region of 180”. Two possible variants of the internal parity of the system (KNY) are considered. The asymptotic amplitudes are represented in the factorized form corresponding to the contribution of two Feynman graphs describing the exchange of a reggeon. The helical amplitudes which are oscillating functions of the energy and angle of production are calculated; the contribution of the longitudinal polarization of a virtual photon proves asymptotically smaller than the transverse polarization contribution. Different observed characteristics of the electroproduction of K mesons are calculated.
1. Introduction The investigation of the analytical properties of partial-wave amplitudes as functions of the angular momentum has proved extremely effective in exploring the asymptotic behaviour of the amplitudes of different processes at high energies ‘). The simplest hypothesis based on the Mandelstam representation and two-particle unitarity has been considered very intensely in quantum field theory. According to this hypothesis the scattering amplitudes at high energies are only determined by the simple poles of the partial-wave amplitude. The investigation of the many-particle terms of the unitarity condition recently *, “) undertaken shows that the analytical properties of partial-wave amplitudes are not as simple as was first thought; in addition to moving poles there are also moving branching points. However, many features of the purely pole asymptotic behaviour hold good in the case of a more complicated picture of analytical properties in the complex j plane. In particular, there remain the spin structure of asymptotic amplitudes, factorizing and isobaric relations and the oscillation behaviour of large angle scattering amplitudes. The production of strange particles in inelastic electron-nucleon scattering at high energies in the angular domain where the asymptotic behaviour of the amplitudes is determined by fermion Regge poles is considered in this paper. The investigation of this process is of double interest; on the one hand, the electroproduction of K mesons is an elementary process of production of strange particles and on the other hand this process possesses some interesting peculiarities making it different from the 637
638
M.
production
of K mesons
P.
REKALO
by real y quanta.
First, the square
of the four-momentum
of a virtual photon is not zero; secondly, besides the transverse polarization the virtual photon has longitudinal polarization; thirdly, the differential cross section for the electroproduction of K mesons averaged over the polarization of all particles involved contains a term corresponding to the photoproduction of K mesons by a linearly
polarized
photon.
by non-polarized
Therefore,
electrons
the investigation
the investigation
at high energies
of the photoproduction
of the production
may furnish
of K mesons
of K mesons
the same information by polarized
as
real y quanta.
2. Structure of the Amplitude in the S-channel The
K
determined
meson-nucleon
electroproduction
amplitude
(e- + N + e- + Y + K)
is
as follows: si,
=
6if_i64(rl+P1-r2-P2-q)
mZ%M2
A,
2wq .QE, E2
(274%
(1)
where Sif is the S-matrix element, rl , m, El (r2, m, E2) are the momentum, mass and energy of the initial (final) electron, pl, M, and E, are the four-momentum, mass and energy of the nucleon, p2, M2 and E, the four-momentum, mass and energy of the hyperon, and 4, w are the momentum and energy of the K meson. If the electromagnetic interaction is taken into account in the lowest order in perturbation theory, we obtain the amplitude
A=
where k2 = (rl -~2)~ interacting particles. The quantity
and J,
is the electromagnetic
EJ is expanded
EJ =
in independently
5Ai(s,t,
~2
current
invariant
operator
of strongly
combinations
k2)Mi(E, k, ~1) p2),
i=l
M,
=
rsR,
M4 = -il3,
M2
=
rEP2,
M,
M,
= TEP1,
s = -(k+pd2,
where r = iys if the internal it is positive.
f2%(P24lJ,lP,h
= -iT&p21,
b!fe = -irEp,&
u = -(k-p2)2,
parity
(2)
of the system
t = -(pi
(KNY)
-P~)~,
is negative
and P = 1 of
639
STRANGE PARTICLES
3. Structure of the Amplitudes in the U-channel; KNY Parity Negative Let us first consider the case of negative parity. Since in the following we shall be interested in the asymptotic behaviour of electroproduction amplitudes, s --) co, u = const, t + -co, which corresponds to the production of K mesons at 180” in the c.m.s. of the K meson and hyperon,
it is necessary
procedure,
in the u channel
the partial-wave
y* is a virtual
photon.
expansions
To realize the transition
make the substitution k -+ -k in eq. (2). In eq. (2) we eliminate the time component conservation equation k,fi(0+,44 and pass amplitude 9,
to two-component
= ia - 8Flu(w,, x F3u(w,,
Then
we obtain
cos
e,) =
87cw,
92u(w,, cos e,)
sqw,,
cos
. ii8 . IW~~(W,,
cos
Thu(wu, cos
ia
. ;E
cos e,)
. LF6u(~,,
amplitudes
cos e,),
as follows:
_M2)_A4],
U
=
e,) = (E,,+M,)J(E,,-M,)(E,“-Mz) 8nw,
~4u(wtl,cos
current
for the two-component
with the invariant
J&.+w)(E2u+m[_qw
to
cos 0,) + ia * 8%~ . tj
+
sqw,,
where
it is necessary
= 0
cos 0,) + ia . @ . Q3F4u(w,, cos Q,) + ia
are connected
+ Y +y*),
eP with the aid of the electron
cos 0,) + ia * ka . EC * tjFzu(w,,
where the scalar amplitudes
by the conventional
(K+N
into the u channel
= Q,,
quantities.
to obtain,
(j ) = (E,,-Ml)~\I(El,+M,)(E,,+M,) u 871~~
0,) =
0,) =
CA3_(W +M2)A6~, U
[-A3_(W
u _M,)‘&],
(E,,-M,)J(E,,+M,)(E,,+Mz) 8j-w,kou
(E2,+M,)J(E,,-M,)(E,,-M,) 8nwu
[_A,cw kou
_M2j_A, u
(3)
640
P. REKALO
M.
The kinematic
coefficients E
here are functions
_ n+W-P2 lu 247
E2u =
’
u-M;-k2 ko, = .It is necessary
of U: tr+M;+k’ 2Ju
w, = Jzl.
2Ju
’
to note that the quantity
k2 (the square of the virtual photon
is assumed to be a finite quantity, which corresponds angles. The invariant amplitudes satisfy under our assumption tion. Hence from eq. (3) follow the symmetry properties 91.(M’,,
= -2
f5 s (-+o~LF.~+o)
the Mandelstam
cos O,),
9-3U(ll?U, cos e,> = -P4u(-M’,,
cos O,),
Let us then introduce
sin 30,
= -2
f6 E (+opF”l*o) = 2 ~0s
helical amplitudes:
Cf:‘(~.)Cpg+;(z.)+p;-~(~.)l,
sin ie,
C~~~(W,)[PJ+)(Z,)+PJ-~(Z,)I,
*e, ~fa’;~.)cp;~:(~,)-P;-~(3.)1. .i
?I = cos e,, connected
with the scalar amplitudes fi = -&/2
by the relations
sin ~eu[2(~,,+~2u)+(i+c~~
f2 = -:JZ cos :e,[2(9-,,-
s2J
e,)(9,,+94Uyj, - (I-
f3 = $42 cos *e,(i - cos e,)(F,,
- Fdu),
f4 = *JZ sin :e,(i + cos e,)(fl,,
+ FJ,
fS = -sin
~eu(~lu-~2u+c0s
f6 = -C~S ~e,(~,,+3=2u+c0s
cos e,w,,-
representa-
(4)
cos 0,) = - P-eu( - W,) cos 0,).
the following
“mass”)
to small electron-scattering
cos Q,) = -92U(-lIt’,,
SJW,)
fi 5 (fllF,l*o)
-’
=cu)it
e,93u-cos
e,~~,+9,,-p,,),
e,9-3u+cos
e,~,,+~,,+9-,,).
STRANGE
If we introduce
the partial
641
PARTICLES
wave amplitudes
with a definite
parity
with the aid of
the relations f/
= i(h( + hi),
f;’ = +(/ii -h$, we obtain according
j-3’ = +(hj+h$
fsj = *(hi + hj,),
fi = #j-h’,),
fJ = +(h< -hi),
to eq. (4) h{(w,,) = -hi(-w,), h’3(WU)= hj4( - WJ,
(6)
h<(w,) = hj6( - WJ. These relations are central in the theory of fermion Regge poles. It follows from them that if, say h{(w,) has a certain singularity in the j plane and its position is determined by the condition j = a(~,), then h{(wJ also has a singularity in the j plane at the point j = a( - w,). If the singularity is a pole, the subtraction of the amplitude h{(w,) at the polej = LY(W,)and the subtraction of the amplitude hi(~v,) at the polej = CX(- wJ are interconnected. Assuming that c((EJ,) is real in the region of bound states, Gribov “) has shown that Im c((w,) # 0 at u < 0 i.e., the amplitudes have an oscillation character at high energies and scattering angle M 180”.
4. Asymptotic
Invariant Amplitudes;
KNY Parity Negative
With the aid of the Sommerfeld-Watson transformation we pass in eq. (5) from summation in j to integrals, deforming the integration contour in the complex region by extending it from a- ice to a+ ice. Assuming that to the right of the straight line Rej = a there lie only poles of partial-wave amplitudes we obtain asymptotically for z, + 00, u = const, corresponding to high energies in the s channel and K meson production angles z 180”, the following expressions, taking into account the conjugation of the trajectories of fermion poles with opposite parity: -+ f1 2 sin +e, ___fi 2 sin :e,
f3 2 cog +e,
j-2 = Q;(&))*
2 cos +e, -
+
~ fi 2 cos :e,
f4
------_=
= -2&Ju)
_2(x;(Ju))*
2 sin +e,
- ~ f4 = -2&/u) 2 sin +e, 2 cos +e, = 2(x:(Ju))* 2 cog 3e,
,i-3f(_s)j-* cos zj
’
sj*-w-#‘-+
)
cos nj*
f3
_+Lf5 2 sin $e,
sj*-++,i._, )
si-3+(_s)i-+, cos nj ~j*-++(-sY*--t cos nj*
)
642
M.
L
fs
-
2 sin *Q,
____
f6
P.
=
REKALO
,j-t+(_,)j-+,
-2&/u)
2 cos +e,
cos rcj
where x1, x2 and x3 are the subtractions II{, h< and hi multiplied by a certain function of u originating in the consideration of the asymptotic behaviour of the Legendre polynomials Using
and the sign (+ ) corresponds
these
expressions
to different
let us determine
the
signatures.
asymptotic
forms
for
invariant
amplitudes Al(s,
u)
=
sj-++p
ul(Ju)
~i*-3f(-sY*-+)
+uy(Ju)
cos 2-cj A,($
+
=
cos
nj*
p1 (I- $) +a,(Ji+M,)] q-~~;y-+
+ [“T (I+ 5) -u;(:u--rM2)]s-y*-+
)
-a,(Ji-M,)
A, =
,j-t*(_,)j-P
sj*-++s)j*-*,
+uT(Ji+M2)
cos nj
A,(s,u) =
(a, +a, Ju
,j-+*(__,)i-+
1 -
cos nj
(7)
cos nj" -
(Ju- 21 *
u;
,j*-l+(_,)j*-t
--a
9
cos nj*
where
‘I (“)
________--4rtJ2 _~ = J(E,,-M,)(E,,-M,)
a2
=
(x: -xl),
4i+k,, (44
JU(E2,+M2)VI(E,.--M,)(E,,~
(xl’
+x:
+JW
The amplitudes A, and A, are asymptotically small. Thus, the invariant amplitudes are determined by the two quantities al and ~2, the asymptotic structure of these amplitudes being such that the total amplitude of the virtual photoproduction of K mesons can be represented as follows: A = A(+)(& u)+P(s,
u),
A(‘)(&u) = u(p,)B[b,(Ju)+il;b,(Ju)](iJ’-Ju)y,u(p,)
,j-+*(_,)j-+
~
cos
A’-‘(S, U) = C(P~)J?[~:(~U) + iLbz(j~)J($+
nj
sj*-++(_s).i*-t Ju)Y~ U(pl) Lyp cos nj*
3
3
(8)
STRANGE
where bl and b, are determined
PARTICLES
by a, and a2 with the aid of the relations b,+b,(&+M,)
2b,-b&h+MJ
= -a,,
= -a,
l(
Eq. (8) corresponds Feynman
643
to the amplitude
3 +a,(Ju+M,). Ju 1
determined
by the contribution
of two pole
graphs. These graphs describe the exchange of a reggeon between the photon
and hyperon, on the one hand, and the K meson and nucleon, on the other; the to the propagation of the reggeon between the quantities ($+J u ) correspond vertices on the graphs, yS corresponds to the K + N -+ reggeon vertex and &(b, + if&,). the reggeon, to the Y +y* vertex. One graph describes the exchange of a reggeon with mass ,/u and spin j(Ju) and the other the exchange of a reggeon with mass -Ju
and spin j*(Ju). 5. Asymptotic
Spiral Amplitudes;
(KNY) Parity Negative
In the calculation of the differential cross sections and the quantities characterizing the different polarization effects in the K meson electroproduction, it is convenient to use the helical amplitudes in the s channel. To calculate these amplitudes in the c.m.s. of the s channel we introduce the two-component amplitude 9,
= ia.&~l,+ib.4u.Eb.12~2,+ib.l;&.q^~3S
+ia.de.4~,,+i6.1;E.f~c~+iu.~&‘1;~6s, where gIs
=
4 El~+“JE2~+!!!,)
[A,(W
s
-M,)-A,],
8?TW,
%F25 =
J(Els-~JE2s-~4
+M,)_A
[_Al(w
4
s
]
9
&CW,
s
3s
=
.
(E2,+~2)~(E1~-“,)02~-M2)~-_~2+(w.+~l)~,], _________
s
87CW,
F4,
=
(E,,-M,)J(E,,+M,)(E,,+M2) LA
+(w
_M
2
I
s
)A
,
5
9
8nw, .F
(E,,-IM,)J(E,,+M~)(E~~+M~) 5,
kos
(El,+M,)4(E,,-M,)(E,,--M,) 6s
+M,)_A s
8nWs
9
C_Al(w
=
[A,(w
_M,)_A, s
= 8nWs
kos
4
(9
644
M.
P.
REKALO
while s+Mf+k2 w, = Js, Using obtain
EI, =
relations
E
2Js
’
2b
(9) and asymptotic
s-M:-k2
= s+M,2-~’ 24s
expression
kos =
)
(7) for invariant
2Js
a
amplitudes
we
for helical amplitudes 1
($)l~sl+l)
=
I
S’--r*(-SY-r
r&/u)
sj*-+*(-s)i*-3)
+(r&/u))*
cos nj (+q~&l_+_l)
=
+(Ju)
cos xj*
si-i~(-4’-t
~j*-%-~~*-3)
+;(&))*
cos nj (g)lgq_g)
=
irF(Ju)
sj-w-4-+
cos nj*
f*-**(-sY*-3) (10)
+:(Ju))*
cos nj*
cos nj ($)l$zT
It-l)
=
@(Ju)
sj-t-*(-s)J-++($(&))*
sj*-w,:‘i”,
cos zj where
r:(Ju) = rz+(Ju)
=
~_ Ju
(xl’+x:+$x:)]
-xl’+xz’-
9
2&4u-Md(E,u-M,) ____Ju
xl’_x;-
2~(-4,-M,)(&,-MJ
2
:M (xl’+x:+$x:)] *
E 2u
2
The helical amplitudes corresponding to the longitudinal polarization of the photon are asymptotically smaller than those corresponding to its transverse polarization and therefore the longitudinal polarization contribution can be neglected.
6. Asymptotic
Invariant Amplitudes;
(KNY) Parity Positive
Let us now consider the case of positive parity of the system (KNY). of the u channel the two-component amplitude has the form Fu
In the c.m.s.
= 0 * La * E9=lu(W, ) cos 0,) + d - 86 * 4^92u(wu ) cos e,> +s * cj~3u(w” ) cos e,>
+ CJ* I;a * $5 . 4LF.+u(w. ) cos e,> + E . dF5u(w, ) cos e,> + d * ka * lj& * w6u(w,
) cos O,),
where 9
= lu
J(Eh‘+Ml)(E2U+M2) 8nw,
[_A,(w
11
+M
)_A 2
] 4
3
(11)
STRANGE
F
3u
PARTICLES
645
d&.+~d(E,u+~,)
=
h43-bJ”--&l~
87CW”
95” =
kuJ(EI”+wE2u+~2) 8nwu kou [A,(w
)_A
_M
4
2
”
+W,A,+E,,A,-W,(W,-M,)A,-E,,(W,-M,)A,I,
9
6u
k,Jch”+wE2U+w
=
+Mz)_A4
[_A,(w u
8nWu
kou
-WuA,-E,,A,-w,(W,+M,)A,-E,,(W,-M,)A,l.
From
relations
(11) there follow 9&(w, 93u(w,,
the symmetry
properties
for scalar
amplitudes
) cos e,> = - 9-2”( - w, ) cos O,), cos e,> = -T&(-w,
) cos S,),
F5u(w, ) cos e,> = - T6”( - W,) cos f3”). Repeating the arguments of the previous sections we obtain the following expressions for the invariant amplitudes in the asymptotic region s -+ co, u = const < 0:
A,(%u>= 4/u> A,(&
u)
=
-
+u;*(~u)~j*-,+(-s)‘*-f
,j-+*(-s)j-+
3
cos
zj
cos nj*
(a;+O;)si-i+(-S~-f ( M, ) Ju 1+
cos nj
A,(s, u) = a;(J;_M,)
l(
3 @;+a;)* 1 44
sj-%(-+Y-*
_a;*(J;+M,)
,j*-Q(-s)i*-+, cos nj* sj”-+*(-Vi
cos nj a;_a; A,(& u) = ___ Ju
,i-+k(_,)j-* cos nj
@)
cos nj* + (a;-a;)* Ju
~~*-+*(-s)j*-+ cos nj*
.
As to A3 and A, they are asymptotically small and a, and a2 are connected with the subtractions of the partial-wave amplitudes. This structure of the amplitudes in the asymptotic region makes it possible to represent the K meson electroproduction amplitude as the sum of contributions of
646
M. P. REKALO
two Feynman
graphs
whose character A =
was discussed
above:
A(+)(& u)+P(s,
u),
sj-++sy--t (13)
A’+‘(s,24)= Zi(p,)&[b;(Ju)+i~b;(Ju)](~-Ju)u(p,)
cos rj
)
,j*-+*(_,)j*-t
A’-+, 24)= u(p,)8[b;*(Ju)+i~b;*(Ju)](lf+Ju)u(p,) while the following
connection
3
cos nj*
holds:
zqju)
= (
b&/n)
1+ 3 +(a; +a;), Jn 1 .
= a!? u
The amplitude (13) in the case of positive parity of the system (KNY) differs from the amplitude (8) in that a unit matrix instead of yS corresponds to the K+N + reggeon vertex. In terms of the internal parity (KNY) this substitution seems quite natural. 7. Asymptotic
Spiral Amplitudes;
(KNY)
Let us calculate the asymptotic helical amplitudes for the positive parity variant (KNY). We proceed plitude
Parity Positive
in the c.m.s. of the s channel from the two-component am-
s +M,)-A,],
Yzs =
4&5+~1)(E2s442)
CA,(w
_M
F3s
=
9sJ(E1s+w)(E2s+~2)
CA
1
4,
1
+(w
2
8nw,
)_A
s
87cw,
_M
s
)A
1
]
5 3
STRANGE
9
5s
kJ&s+wEzs+M2)
=
Bnw,
du-
6s
PARTICLES
647
[Al(w
s
_M
kJ@ls-w)(E2s-w [_qw
=
1
)_A
4
kos
)_A
+M
s
1
4
8nwskos
-E,,A,-~,A~+E~~(w,-M~)A~+wS(W,+M~)A~I. Using the asymptotic amplitudes
@)jgsl
-3_
1) =
invariant amplitudes (13) we then obtain for the helical
_
$(Ju)
sj-+;+y-+
sj*-+i(-f*-’ ,
++;(Ju))*
cos (g)]
+~sl_+l)
=
+gs,+_l)
=
p;(JU)Sj-++wf
(15)
sj*-++w-,
_ir~(Ju)s’-+f(-sY-k++.;(&))* cos nj
($0,
7q
cos nj*
si*-3+c-r+)
+($(&))*
cos rcj
cos nj*
where r1 and r2 are connected as follows with the quantities a; and ai determining the factorized amplitudes r,(Ju)
= -I!!!5
r,(Ju)
= - -Yf!f-
G-al
8rrJ2
, Ju+M,
ju_M,
T
8x,/2
The helical amplitudes corresponding to the longitudinal polarizations of a photon are asymptotically small. 8. Characteristics
of Electroproduction
Using eqs. (10) and (15), we can determine different characteristics of the electroproduction of strange particles e.g., cross section, polarization hyperons and asymmetry. To this end let us first re-write eq. (1) as
si, =
e
2.Y
kZ (2~)~
2kos
---A(y*+N e1c2
-+ Y+K),
(16)
where A (Y*+ N + Y + K) is the matrix element of virtual photoproduction in which the quantity Ey,u gives the polarization and the factor J2k,, originates from the normalization.
M.
648
Then
the differential
can be represented do
=
cross section
REKALO
for the electroproduction
of strange
particles
as follows: (kp,)* + k*Ivff
a ko,m2 rlpl TC*elc2k4
where J(r,
P.
(rIpI)*-m*Mf
kp,
p2)* - m2 Mf/r,p,
da(y* + N + Y + K)d3r2,
is the relative velocity
of the initial electron
(17)
and the
of the nucleon of the target and J(kp,)*+k*Mf/kp, is the relative virtual photon and nucleon introduced in order to go over from (A(y* + N -+ Y + K)I* “velocity”
to da(y*+N -+ Y+K). After the averaging over the initial
and summing
over the final electron
spin states
we obtain do(y*+N
--f Y+K)
= 4& s
s,fi f,s don,
where I%’is the unit vector orthogonal to k and located in the plane produced by the vectors k, rl. In eq. (18) the term J,,f,, corresponds to the differential photoproduction cross section averaged over the virtual photon polarization. The term d~&J& corresponds to the cross section for the production of K mesons by transverse linearly polarized photons, 6,&j& is responsible for the production of K mesons by longitudinally polarized virtual photons, and (k,& + I%:k,)f,, describes the contribution due to the interference of the amplitudes for the production of K mesons by transverse and longitudinal virtual photons. Since the contribution from the longitudinal polarization of photons is asymptotically small, eq. (18) can be re-written as da(y*+N
-+ Y+K)/dQ
k2 de = smz dn +
(19
are the differential cross sections for the production where do/dQ and (da/dQ)(e,) of K mesons by non-polarized and linearly polarized transverse photons, and the axis x is directed along the vector k’. For the polarization of recoil hyperon we obtain in a similar manner p da(y*+N
+ Y+K) dQ
where P’ is the polarization vector when the photons the same vector when the photons are linearly polarized.
are non-polarized
and
P”
STRANGE
649
PARTICLES
For the azimuthal asymmetry of the production of K mesons on polarized nucleons, we have A do(y*+N
+ Y+K) dS2
9. Asymptotic Expressions For Characteristics of Electroproduction of Strange Particles Let us give the explicit asymptotic expressions for different quantities in eqs. (19)-(21), considering simultaneously both internal parity variants: do = ~(1+~:)(p12+p:2)s2j’-1, dS2 -
gje.) = g
+cos @(l-c&) cos (q,: -cp:)p:p;s2”-l,
(22)
where r:(Ju)
=
pfP’*,
r;(~u)
=
P;eiq2*,
a: =
chd’+slw’ ch nj” T sin 7cj’
and @ is the azimuthal angle. These quantities do not depend on the internal parity (KNY). For the polarization of recoil hyperons we have
P;!ti(e,) = cos@P;.g -cos@R, -COS
Py g
(e,) = sin @R, -sin3@R,,
3@R,,
P;!= 0,
where
R, = *(cc, sin fl)~:~s’j’-~,
R, = $(a,sin fi)p:‘s2”-r,
tg/?=-.
sh nj” cos ?rj’
The plus sign in [ &-] corresponds to the negative parity of the system KNY and the minus sign to the positive parity. The system of axes x’, y’, z’ is connected with the recoil hyperon, being directed along the momentum of the hyperon.
650
M.
The asymmetry has the following
of the production
K.
REKALO
of K mesons
form when the photons
on polarized
nucleons
of the target
are non-polarized:
A;$ = t(l+~“,)(p:“+p:z)s2j’-l,
and the following
do A: dn = [i-]a+
sin /?(p:2+pt2)
sin @s2j’-i,
Ai g
sin /I(pf2+pi2)
cos @s2j’-i,
= [T]a*
form when the photons
are polarized:
A:’ $
(e,) = Ai g
+(sin
Ai’ $
(e,) = ‘4; $
+ (cos @ - cos 3@)C,
C = [-k]x*
sin bp:pz
@+sin 3@)C,
sin (cpf -‘p:)s2j’-l.
Contrary to the oscillatory behaviour of helical amplitudes all the above quantities are monotonic functions of the energy and angle. Only those quantities which characterize the polarization of recoil hyperons versus the polarization of the target nucleons are oscillatory, e.g. TzY,$ Q, = [r]+(~+
(4
=
Q1 +Q2,
Q2 = -4 h (fisfZse-iQ)+f3*sf4se3iQ),
sin p)(p:2+p~2)s2j’-1,
Im_flsf,*, = Imf3J4*, = -C, Refdi*,
= [&IP:P:[c&
sin (j”i+qp1kP)
cos (j”i+& -sin
Re_LJ,*,
=
[+ldd[ cc+ ? sin
(j”i+qP:*B)
cos (j”~+~P:)]s2j’-r,
cos (j”i+cp:+fl) -sin
where i = In sJsO and s,, is a certain squared.
(j”[+qi)
fP)
quantity
(j”[ + cpi) cos (j”c+ with the dimension
(p2+)]szi’-r, of an energy
10. Conclusion In conclusion let us formulate briefly the main results of this paper. (1) The amplitude of electroproduction of strange particles for asymptotically high energies and K meson production angles w 180” in the c.m.s. of final strongly
STRANGE
interacting
particles
can be represented
651
PARTICLES
in the factorized
form (8) or (13) depending
on the internal parity (KNY). This structure of the amplitude corresponds to the specific Feynman graphs describing the exchange of a reggeon. (2) The helical amplitudes in the asymptotic region are oscillating functions of the energy and angle; these amplitudes have a simple structure, which facilitates calculation of different characteristics of the process under study. (3) The contribution
of longitudinal
photons
is asymptotically
smaller
than
the the
contribution of transverse virtual photons. (4) Only the polarizations of recoil hyperons versus the polarization of target nucleons are observed oscillating quantities. All the other quantities are monotonic functions of the energy and angle. The author results.
wishes to thank A. I. Akhiezer
and D. V. Volkov for discussion
of the
References 1) V. N. Gribov, JETP 41 (1961) 66, 1962; S. C. Frautschi, M. Gell-Mann and F. Zachariasen, Phys. Rev. 126 (1962) 2204; S. C. Frautschi, G. F. Chew and S. Mandelstam, Phys. Rev. 126 (1962) 1202 2) S. Mandelstam, Nuovo Cim. 30 (1963) 1113, 1127, 1148 3) V. N. Gribov, I. Ya. Pomeranchuk and K. A. Ter-Martirosyan, Preprint, No. 222 Inst. for Theor. and Exp. Physics 4) V. N. Gribov, JETP 43 (1962) 1529